In this note, we study automorphic forms and representations of . First, we describe local theory, archimedean and non-archimedean, and then global theory. This note is mainly a summary of a part of Bump’s Automorphic forms and representations (bu?), from chapter 2 to 4.

Introduction

Modular forms and Maass wave forms are certain functions defined on the complex upper half plane that satisfies -transformations laws (or more generally, transform under congruence subgroups ). There are a lot of applications of modular forms in number theory, such as sum of squares and the irrationality of , and the Wiles’ famous proof of Fermat’s Last Theorem. There are also applications in other subjects, such as combinatorics (partition numbers), physics, representation theory (monstrous moonshine), knot theory, etc.

In this note, we will study how to interpret such functions (so-called classical automorphic forms) as a representation of adéle groups (here is a ring of adéles of global fields such as ), and study representation theory of it. This can be a starting point of the Langlands’ Program, which connects representation of Galois groups, algebraic geometry, and automorphic forms (representations).

To study such representations, we first study local representations. There are two kinds of local representations - archimedean and non-archimedean. For the archimedean cases, we study representation theory of via so-called ()-modules. -module is a vector space with compatible and -actions. It is easier to study -modules than studying the representation of directly since -modules are more algebraic. We will classify -modules for and also study which of them are unitarizable, since we are interested in the representation that lives in space. Also, we will see how these representations are related to classical automorphic forms (such as modular forms and Maass wave forms).

We also have non-archimedean representations - which are representation of -adic groups for a prime . They are very different from archimedean cases because of their topology. This makes the situation easier or harder, but anyway, we will also classify all the representations of such groups and study their unitarizability.

When we finish the local theories, we can glue these representations to obtain the representation of the adéle group . (In fact, this is not a true representation of , but a representation of .) While we are studying such representations (local or global), we will only concentrate on some nice representations (admissible representations) that are close to the representation of finite groups. Automorphic representations are some nice representations that also satisfies some analytic conditions on growth. Later, we will see that Flath’s decomposition theorem tells us that it is enough to study such glued representations to study automorphic representations.

Before we get into the representation theory of , we will study first, which are completed by Tate in his celebrated thesis. He find a natural way to prove the analytic continuation and the functional equation of Hecke’s -function using local-global principle, and such idea will be used to define -functions attached to automorphic representations of .

It may be hard to study an abstract representation of a given group (such as or ). Whittaker model (or Whittaker functional) help us to study such representations as a very concrete representation that functions on the group lives (and the group acts as a right translation). Most case, such Whittaker model exist and unique, and such results are called (local or global) multiplicity one theorem. In the last section, we will see how the multiplicity one theorem is related to the classical modular forms.

Archimedean theory

In this section, we will study representation theory of the group . Usually, it is easier to study representation of compact groups than non-compact groups because it is not much different from the representation theory of finite groups. First, any finite dimensional representations are unitarizable, by taking average of arbitrary hermitian inner product on the space over all group with respect to Haar measure, which is finite for compact groups. Also, we have celebrated Peter-Weyl theorem, which claims that any unitary representation (including infinite dimensional representation) on a complex Hilbert space is semisimple, i.e. can be decomposed as a direct sum of irreducible dimensional unitary representations, and these are all finite dimensional and mutually orthogonal. It is also known that representation of compact group are completely determined by its character.

Also, Lie algebra representations of (or its complexification ) are much easier than studying the representation of Lie group, because it is a linearlized version of original representation and we have a lot of tools to use. We even have a complete classification of semisimple Lie algebra over , which is a very rich theory itself.

Instead of studying representations of directly, we will study representation theory of its maximal compact group and Lie algebra representation of . Eventually, we will consider so-called -module, which is a vector space with compatible actions of and , and the space is not so big to deal with, i.e. admissible. We give complete classification of -module for and , and investigate which of them are unitarizable. Since unitary representation of is completely determined by associated -module, we also get a complete classification of unitary representations.

In the last subsection, we will also see how the representation theory of can be used to study spectral problems (of classical automorphic forms).

Representation theory of

Geometrically, Lie algebra of a Lie group is a tangent space at the identity, and it has a structure of Lie algebra given by a Lie bracket. In case of and , their Lie algebra is , the space of real matrices with the Lie bracket . The most important point is that any representation of Lie group induces a Lie algebra representation.

The finite dimensionality assumption is non really necessary. In fact, we will only consider special kind of representation: right regular representation on . The statement is also true for this case, even if the space is not finite dimensional.

By the universal property of universal enveloping algebra , any Lie algebra representation can be extended to a representation of . We will regard as a ring of differential operators, which are left-invariant since Lie algebra action is obtained by differentiating right regular representation. When the element is in the center of the universal enveloping algebra , it is both invariant under the left and right regular representations.

Now we will concentrate on . is generated by the elements with relations Now let be an element in , where the multiplication is in , not a matrix multiplication. This is a very special element in , which is called the Cacimir element. The element is in the center of , and in fact the center is generated by and .

We will consider the complexification of and slightly modify the elements in as Then they satisfy the same relations as , and . Indeed, we have where is the Cayley transform. We will see the reason why we are using instead of , and , in section 2.6.

For an arbitrary representation of , there may not exists a corresponding Lie algebra action on since the limit may not exists. We will define as a largest subspace where such action exists, i.e. the limit exists for all and . We will call such as smooth vector, and we can easily check that such space is invariant under the action of from the equation Also, the action of on is a Lie algebra representation. We define the action of on as for . We can show that the subspace of smooth vectors is not so small, indeed, it is dense in .

Representation theory of compact group

In this section, we will see how representations of compact groups well-behaves. We will prove the Peter-Weyl theorem, which claims that every representation of a compact group decomposes as a direct sum of finite dimensional irreducible representations.

For any finite group and it’s irreducible representation (which has finite degree), we can construct a -invariant inner product on : choose any inner product and define a new pairing as Then this pairing is also an inner product on and it is -invariant by definition. We can do the same thing for a representation of compact group on a Hilbert space , by integrating a given inner product on over with respect to its Haar measure. (Note that compact group has a finite Haar measure.) This induces same topology as before.

Now we will prove the most important theorem in the representation theory of compact groups, Peter-Weyl theorem. For a representation on a Hilbert space of , a matrix coefficient of the representation is a function on of the form . We need the following proposition:

Using Peter-Weyl theorem, we can define admissibility of representation of for or . A representation of is admissible if each isomorphism class of finite dimensional representations of occurs only finitely many times in a decomposition of . This implies that for each irreducible representation of , the isotypic component of , the direct sum of all the subrepresentations of isomorphic to , is finite dimensional. We can check that multiplicity of a given finite dimensional representation does not depend on the decomposition. Also, it is a right category to study since it is known that any irreducible unitary representation is admissible.

The next result shows that in the decomposition of irreducible admissible unitary representation over , the multiplicity of the trivial representation of is at most one. To prove this, we need the result about commutativity of Hecke algebra which can be proved by Gelfand’s trick with Cartan decomposition.

Note that is non-commutative.

For , we can prove a similar result when we consider the subalgebra of where acts as a nontrivial character , i.e. . Let be a subalgebra of such functions.

Now we can prove the uniqueness of the -fixed vector.

Note that the admissibility condition is unnecessary because any irreducible unitary representation is admissible (as we mentioned above).

-module for and classification

Now we can define the -module, which is a thing what we really want to study. In some sense, the subspace of smooth vectors is still too big to study. We will consider much smaller space, the space of -finite vectors , which is also dense in but much easier to study algebraically.

Motivated by this, we define a notion of -module, which is a vector space of -finite vectors with compatible actions.

For example, if is an admissible representation of , then is a -module. We will classify all the irreducible admissible -module for . First, we will do for with , and modify it to get the result for with .

Let be a irreducible admissible -module, so that it can be decomposed as an algebraic sum of isotypic parts where each is finite dimensional. Since is abelian, all the irreducible representations are 1-dimensional, and they are parametrized by integers as . Hence we can write as where . Each is at most 1-dimensional by Theorem . Now we can extend the -action to -action naturally. The set is called the set of -types. We have the following Schur’s lemma for -modules.

This proposition shows that the elements acts as scalars on . (This will be the parameter to classify -modules later.) The following proposition gives a description how the elements acts .

By the proposition, we have that the set of -types of is all even or all odd. This defines a parity of , even or odd. The following theorem tells us the uniqueness of representations, whether we don’t know the existence yet.

Now we will give a construction of such representation with given parameters, which will finish the classification. Let or , which represents parity of a representation, and let be two complex numbers. Let and , where . As you expect, these will be scalars corresponding to and .

Note that the right translation action (regular action) extends to . We can also prove that is the space of smooth vectors for this representation. By Iwasawa decomposition, we have for , so each is determined by its value on , and can be any smooth function, subject to the condition .

In fact, the representation is an example of an example of induced representation. For a locally compact Hausdorff group and its subgroup , we can obtain a representation of from a representation of in a canonical way: if is a representation of , then define where are modular characters. If we give -action on by right translation, then this gives a representation of . We denote such representation by . Now let , (the subgroup of upper triangular matrices in ), and let be a character defined as Then, by definition, the representation is just . Note that is a unimodular group (so that is trivial) and

Now we want to study -module of -finite vectors in . If , then there exists a unique such that , which satisfies . Iwasawa decomposition gives an explicit description of : By the direct computation, we can show that this function satisfies the following relations:

Now, as you expect, these representations give examples of the previous representations with two parameters and -type. The above ’s generate the space of -finite vectors.

In other words, this gives a classification of -module for .

When is not of the form , then the equivalence class of irreducible admissible -modules of with given are denoted by . When , we denote it as and it is called principal series. (By tensoring with a suitable power of determinant, we can assume easily.) Later, we wiil check that the representation is unitarizable if and only if and , so we will concentrate on this case.

The finite dimensional representation with a set of -types can be realized as a space of polynomials: consider the space of homogeneous polynomials of degree in two variables , and let acts on the space by which is a degree irreducible admissible representation where acts as the scalar . This will not appear again since it is not unitarizable. (We will prove that the only finite dimensional unitarizable representation is 1-dimensional, which factors through the determinant map.)

If , we have irreducible admissible representations with set of -types as , and equivalence class of these representations will be denoted as and called the discrete series. When , the representations are called limit of discrete series.

To classify -modules of , we need some modification. We can check that the representation of has a symmetric property: the set of -types is symmetric so that if and only if . Hence cannot be extended to , but can be. We will denote the latter one by , with -module structure.

For the construction of principal series reresentation of , we define as for , where and . Then we denote as , and we will denote by the underlying -module of -finite vectors. Note that since each function in is determined by its restriction to . So there are two extensions of the -module structure on to a -module structure (corresponds to the choice of ), and the same is true for the corresponding -modules.

Unitaricity and intertwining integrals

Now we will see which representations in the above list are unitarizable. For some special case (so-called complementary series), we will show that the representation is unitary by using the intertwining integral, which is an hidden explicit isomorphism between two isomorphic -modules.

The following theorem tells us that induced representation of unitary representation is again unitary in some special case.

Using this, we can prove that there are some class of representations that are induced from unitary representation, so is unitarizable.

If is a unitary representation of and , then the action of on is skew-symmetric, i.e. for all . Especially, we have . The following theorem give some necessary conditions for unitaricity.

From the above theorems, we know unitarizability of except for and . We will also show that these representations are also unitary, but induced from nonunitary representations of Borel subgroup. Such representations are called complementary series, and the corresponding eigenvalues are exceptional eigenvalues.

To construct such representation, we will use intertwining integral. We know that and are isomorphic, when they are irreducible, as -module since they have the same and . Also, when they are not irreducible (when ) they are not isomorphic, but their composition factors are isomorphic. We will construct an intertwining map between those to representations as an integral.

For , the operators are defined by The next proposition shows that this is the desired intertwining map, when the integral converges.

We can also compute the effect of on a -finite vector.

Now we can prove that the complementary series is unitary.

In contrast, finite dimensional representations are not unitary in general. In fact, the easiest ones are the only one which are unitary.

The only thing remain that we have to figure out is unitarizability of discrete series. We will prove that there is a unitary representation in the infinitesimal equivalence class for , by constructing such representation on a space of holomorphic functions on which has bounded -norm. We know that if the representation is unitary, and we may assume as before.

The limits of the discrete series representation also can be realized in a space of holomorphic functions on with the norm but we don’t need this since they are subrepresentations of , which is already unitary.

Until now, we studied which -module arises from irreducible admissible unitary representation of . The following theorem tells us that this actually classifies all the irreducible unitary representations.

By combining all of the results, we get the following classification.

Whittaker models

Now we know all the representations of . However, if someone give an arbitrary abstract representation, then it is not easy to study it directly. To resolve such a problem, we may realize the abstract representation as a space of certain functions with an explicit and easy action (right translation). This is a main philosophy of Whittaker models, and we will show that it is possible to realize almost all representations as a space of such functions.

Let be a smooth function that satisfies for a fixed nontrivial unitary additive character of , which has a form of where . We say that is of moderate growth if, when we express the function in terms of via Iwasawa decomposition, it is bounded by a polynomial in as . We say that is rapidly decreasing if as for any . We say that is analytic if it is locally given by a convergent power series. The function satisfying the functional equation and of moderate growth is called Whittaker function.

The following proposition shows uniqueness of such function with fixed eigenvalues of and .

From this, we can prove uniqueness of Whittaker model.

Such uniqueness is important, and we also have uniqueness theorem for non-archimedean local fields (we will prove this in Chapter 3 using the theory of Jacquet functor). By combining uniqueness result for archimedean and non-archimedean local fields, we get the global result, which is called multiplicity one.

Classical Automorphic Forms and Spectral Problem

In this section, we will see how the representation theory relates to classical modular forms, Maass forms and spectral problems.

First, the elements coincide with the classical Maass operators. Recall that we have (weight ) Maass differential operators and the (weight ) Laplacian which acts on the space of smooth functions on , the complex upper half plane. Since , we can lift such function as a smooth function on , so on by letting it translation invariant under . This gives a 1-1 correspondence between space of functions on and on . More precisely:

The main point is that under this isomorphism, Maass differential operators and the (weight ) Laplacian operator correspond to the elements we defined.

We know that there are three types of automorphic forms on :

1. Holomorphic modular forms: For a given character and with , a weight holomorphic modular form on is a holomorphic function satisfying for all , and holomorphic at the cusps of .

2. Maass forms: Also a function on , but smooth, not holomorphic. satisfies for , and an eigenfunction of the Laplacian operator .

3. The constant function. for all is clearly invariant under (any) discrete subgroup . More generally, is also an automorphic form for any .

Why are there precisely these types of automorphic forms on and no others? You may see that this list of automorphic forms are very similar to the classification of -modules of (and ). In fact, this gives an answer to the above question. We can consider such an automorphic form on as a function on (by the above map ), and we can consider a -submodule generated by the single element . This is an irreducible admissible -module (admissibility is a result of Harish-Chandra, see Theorem ), and the previous classification gives us three types of automorphic forms.

Another important question is the spectral problem. We can formulate it as follows:

1. Determine the spectrum of the symmetric unbounded operator on .

2. Determine the decomposition of the Hilbert space into irreducible subspaces.

We don’t know the complete answer yet, but we understand some of them. First, one can prove that such decomposition exists.

Now the previous classification of irreducible admissible unitary representations of gives the decomposition of . For each irreducible subspace of it, acts as a scalar on and it depends only on the isomorphism class of . According to the value of , the different types of irreducible admissible unitary representations occur as constituents of the decomposition with some multiplicity.

It is not hard to compute (using Riemann-Roch theorem or other tools), but it is extremely hard to compute and we conjecture that all of them are one, but until now, we don’t know any single exact value of it. (There are some known upper bounds.) It is known that if is cocompact (i.e. is compact), it is known that the spectrum of on is discrete and the eigenvalues tend to infinity.

Non-archimedean theory

Now we will get into the representation theory of over non-archimedean local fields. Archimedean and non-archimedean cases are very similar, but also very different. Their topologies are completely different from archimedean case, which make the situations easier or harder. However, their representations are very similar. For example, we can construct most of the representation from principal series representations, which are induced representations of characters of Borel subgroup, as in the archimedean case.

There are some other representations that do not come from principal series representations, which are called supercuspidal representations. Such representations are also interesting, and we will present some methods to construct such representations (Weil representations).

Smooth and admissible representation

In this section, we will fix some notations as follows:

• : a non-archimedean local field

• : a ring of integers

• : the unique maximal ideal of

• : a uniformizer, i.e. generator of

• : a residue field

• : cardinality of

• : normalized valuation of

• : nomalized additive Haar measure

• : normalized multiplicative Haar measure

The biggest difference between archimedean and non-archimedean local fields is the topology. Every group over non-archimedean local fields that we will see will be totally disconnected locally compact spaces. Such groups always have a basis of open subgroups at the identity, which can be chosen as normal subgroups when is compact. For example, in case of , the subgroups () of elements in congruent to identity modulo forms such a basis, and these are even normal in a compact subgroup .

As in the archimedean case, we will concentrate on representations that we can handle, which are smooth and admissible representations.

One can check that complex representation of is smooth if and only if the map is continuous, where is given by usual complex topology.

Admissible representations are important because they satisfy important properties that also holds for representations of finite groups. Also, most of the properties can be proved by using the corresponding result of representations of finite groups. For example, the following theorem shows that any smooth representation of totally disconnected locally compact group is semisimple, and each isotypic part of the decomposition is finite dimensional if and only if the representation is admissible.

We call that a linear functional is smooth if there exists an open neighborhood of identity such that for all and . We will denote the space of smooth linear functionals as . For any representation , we define its contragredient representation by By smoothness, we can check that also decomposes as so the contragredient of an admissible representation is admissible.

As in the archimedean case, representation of on the space induces an action of Hecke algebra of compactly supported smooth functions, i.e. locally constant functions, where the multiplication is given by the convolution The action of is given by which satisfies . Note that the above integration is actually a finite sum. There’s no identity in the algebra . However, for any compact open subgroup of , the subalgebra of -biinvariant functions has an identity element The following proposition shows that irreducibility of the representation is equivalent to irreducibility of correponding Hecke algebra representation.

Another important feature is that irreducible admissible representations are determined by their characters. For a representation of a finite group , we defined its character as a trace of the representation, i.e. the function defined as . We can also define character of admissible representations as a distribution on . The key property of trace is that the trace of is same as the trace of restriction on any invariant subspace . From this, we can define the character as follows: for any , there exists an open compact subgroup such that . Then is invariant under , which is a finite dimensional subspace, so the trace of the map is well-defined and we let .

Using the theorem, we can prove that contragredient representation of is isomorphic to other representations on the original space with different actions by comparing characters.

Here the central character is a character corresponds to the action of restricted to . Note that the center acts as a scalar by Schur’s lemma.

By the previous theorem, we can directly check that irreducibility of admissible representation is preserved by taking dual.

There’s one more thing worth to mention about totally disconnected locally compact groups. We use the following no small subgroup argument several times, which is very useful and important.

Distributions

In this section, we will briefly introduce properties about distributions that will be used in the later chapters. For a totally disconnected locally compact space , we define a distribution on as a linear functional on . Note that there’s no restriction that the functional is continuous. We denote for the space of distributions on . We have an exact sequence:

We can also define actions of on , and by left and right translations. More precisely, we have for and . The following proposition shows that a distribution which is left -invariant up to some character of is unique up to constant. The proof is not so hard, but this proposition will be used a lot later.

Finally, we introduce the concept of cosmooth modules and there relation with sheaves on .

The following proposition shows an equivalence of category of sheaves of -modules and the category of cosmooth modules over .

The following theorem of Bernstein-Zelevinsky will be used in the proof of the uniqueness of local Whittaker models, i.e. local multiplicity one theorem.

Here is a space of -valued distribution, which is a space of linear functional on . In the proof of local multiplicity one theorem, this will help us to prove certain distribution is zero by only proving it fiberwise.

Whittaker functionals and Jacquet functor

Like archimedean cases, non-archimedean theory also has a notion of Whittaker models. Let be a nontrivial additive character of and be a character of , the group of upper triangular unipotent matrices in , by

The following theorem claims that local Whittaker functional is unique (up to constant), which is referred as local multiplicity one theorem.

For the proof, we need a lemma, which claims that a distribution that transforms like Whittaker functional under left and right translations (we will call such distribution as Whittaker distribution, only in this note) is invariant under certain involution on the space of distributions. Let be a distribution. is called a Whittaker distribution if for all . We define an involution by , where This also induces an action on and . Note that .

Since we just proved uniqueness, we wonder about existence. We will prove that any irreducible representation of of has a Whittaker model. For this, we need a concept of (twisted) Jacquet functor.

We can also define the twisted version of the Jacquet functor.

First important property of these functors is exactness.

Another important property of the twisted Jacquet module is that it is directly related to the space of Whittaker functionals. This is almost direct from the definition.

Now we will prove our main result - existence of a Whittaker functional. For a -module , we can associate sheaf of -module by defining the -action as (Here we fix a nonzero additive character .) One can check that is a cosmooth -module under this action, so we have a sheaf associated to . Using this, we can prove our theorem.

Note that this is not true for , but it is still true for generic representations in the sense of Gelfand-Kirillov dimension.

Like archimedean case, we can also think the Whittaker functional as a Whittaker model, which gives a concrete model of a given representation. This is almost same as the archimedean case. In non-archimedean case, we also have Kirillov model, which is a model given by functions on .

Note that if Whittaker functional is nonzero, then the Kirillov model is also nonzero. (For the proof, see Proposition 4.4.6 and 4.4.7 in (bu?).) It is not easy to describe the action of on , but the action of is rather easy: With the action of center by central character, this completely determines the action of . We will investigate more explicitly in the later chapter.

Another important property of Jacquet functor is that it sends an admissible representation to an admissible representation. Proof uses Iwahori subgroups and the Iwahori factorization. See page 466–469 of (bu?) for the proof.

Classification

Now we introduce the classification of irreducible admissible representations of . Definition of each terms will be defined in following sections.

Classification of representation of (over a finite field) is almost same. For details, see chapter 4.1 of (bu?). For the later chapters, we will study about these representations.

Later, we will see that global automorphic representations decomposes as a product of local representations. For almost all place , the -part of the representation will be spherical principal series representation, which corresponds to certain characters . The classification of unitarizable principal series representation is somehow similar to the archimedean theory.

Principal series representations

In short, principal series representations are representations induced by characters of Borel subgroup (same as archimedean case). Usually, they are irreducible, but there are some special cases that the induced representations are not irreducible. In that case, it has an infinite dimensional irreducible subrepresentation (or quotient) with 1-dimensional complement. Such infinite dimensional representation is called (twisted) Steinberg representations.

The definition of induced representation is slightly different from that for finite groups. We need some extra factors for latter purpose. We did the same thing for archimedean case (see Section 2.4).

By definition, (compact) induced representations are also smooth. As before, we have Frobenius reciprocity, which we need extra factor again.

Now we can define principal series representation. (Compare this with the archimedean case in Chapter 2.4.)

We can also consider the compact induction . In this case, they agree since is cocompact by the following theorem:

Now we can ask some basic questions about principal series representations:

• When is irreducible?

• What is a contragredient representation of given principal series representation?

• When two principal series representations are isomorphic?

• What is a Jacquet module of it?

Note that does not imply that is irreducible, since the representation may not be unitary. So we need other approach to prove irreducibility. We will use Jacquet functor that we defined in the previous section.

First, we will prove uniqueness of Whittaker functional, and use it to prove irreducibility. The argument is similar to the proof Theorem . (We can’t use the uniqueness result in the previous section since we don’t know whether the representation is irreducible or not.)

Let’s answer the second question first. It is relatively easy to answer the second question, and we will use this for the first question.

Now we return to the first question. The following lemma proves that if has a 1-dimensional subrepresentation (or quotient), then should satisfy some relation.

Now we can examine when is irreducible.

When it is irreducible, we denote the isomorphism class as . If it is reducible, we showed that it has two composition factors in its Jordan-Hölder series, a 1-dimensional factor and an infinite dimensionalfactor. In either case, the infinite dimensional factor is irreducible and we denote its isomorphism class as . (Irreducibility of can be proved by using again. If it is not irreducible, we may assume and has a 1-dimensional subrepresentation. From this, we get a 2-dimensional subrepresentation of . However, one can show that every finite dimensional representation of factors through the determinant by no small subgroup argument, and proves that such representation should be 1-dimensional. In other words, the only finite dimensional representation of is 1-dimensional.) Such representation is called a special or Steinberg representation. The 1-dimensional factor is denoted again. There are also other kinds of representations - supercuspidal representations - which we will see later.

Now, let’s answer the next question. When two principal series representations are isomorphic?

We can even write the isomorphism explicitly via intertwining integral. We will define such map as an integral Formally, this gives an intertwining map since holds and where the third equality follows from the substitution . Also, if is locally constant then is also locally constant. At last, is a nonzero map since the function is in and satisfies .

However, the integral may not converges. The integral converges when satisfies certain relations: if we fix two unitary characters and take for , then the integral converges when .

By interchanging roles of and , we also get a map for . So we still have a remaining case of , which is the most interesting case since it is conjectured that the only when can occur as a constituent in an automorphic cuspidal representation. To extend the map for this case, we will analytically continue the map with respect to the complex parameters . There will be a pole when , but we don’t need to worry about this case since we automatically have .

Let be the space of functions on that satisfies for all . For each and , there exists a unique extension of to . We will refer to as a flat section of the family .

The composition is a scalar by Schur’s lemma because is irreducible for most . We can compute this scalar, and it is given by product of two gamma factors. (For the definition of the Tate gamma factor, see Chapter 4.1.)

This computation is useful for the functional equations of Eisenstein series. Also, Kazhdan-Petterson, Bank showed that the image of intertwining integral is irreducible by using this. Also, this helped Harish-Chandra to compute the Plancherel measure of .

At last, we can compute Jacquet module of principal series representations. More precisely, we can compute the action of on explicitly. First, we have a classification of 2-dimensional smooth representations of .

Supercuspidal and Weil representations

We just saw principal series representations and special representations. One can show that if is an irreducible admissible representation of , then . (See Proposition .) We also know that , and it is known that the converse is true: any irreducible admissible representation with is a principal series representation. Also, using the exactness of Jacquet functor we can prove that for 1-dimensional representions or special representations (twisted Steinberg representations). So we have only one more case left: when .

Such representations don’t come from principal series representations, and they are interesting itself. One way to construct such representation is using representation of . We also have a similar classification of irreducible representations of the finite group , and there are so-called cuspidal representations of , which are representations that there’s no nonzero linear functional invariant under . (For the detailed explanation about representations of , see the chapter 4.1 of (bu?).) One can get supercuspidal representations of by using cuspidal representations of .

There’s direct way to construct such representations by means of Weil representations. Here we assume that the characteristic of is not 2.

To get a true representation of (and ), we need to lift the projective representation . We can interpret projective representations as a cohomology class in (or if the representation is unitary), and we can show that when is even, the the corresponding cohomology class vanishes so the projective representations can be lifted to a true representation.

The proof uses quaternion algebra and Hilbert symbol. Note that this is false for even dimension, and the corresponding cohomology class of defines an important central extension of called the metaplectic group.

Using this, we can construct a supercuspidal representation of . Let be the representation of , and let be the restriction of to . Howe conjectured that there’s certain duality between representations of and .

We are interested in rather than , and it is possible to modify Howe duality as a correspondence between representations of and , the group of automorphisms of that preserves up to constant.

When . In this case, the quadratic space can be identified with , where is a 2-dimensional commutative semisimple algebra over and is the norm map. We have two possible cases: when splits ( and ) or not ( is a quadratic extension and ). We can embed into by , and we also have a nontrival involution . Those two generates subject to the relation and .

Now let be a quasicharacter. It is known that can’t be extended to the quasicharacter of if and only if does not factor through the norm map . (This follows from Hilbert’s theorem 90.) In this case, we get an induced representation of , and there exists a corresponding representation of under the theta correspondence. This gives a supercuspidal representation. For non-split case, such representation can be described as follows:

When , similar construction gives Whttaker models for principal series representations.

Spherical representation and Unitarizability

In Chapter 3, we will show that every automorphic representation of (we will define this in Chapter 3) decomposes as a restricted product of local factors, where almost all are spherical (or unramified) representations. Spherical representations is defined in the following way.

We can show that dual of spherical representations are also spherical.

One of our aim in this section is to understand the structure of the spherical Hecke algebra . First, we show that this is commutative, and the proof is almost same as archimedean case (Theorem ), which uses Cartan decomposition and Gelfand’s trick.

Now let be an irreducible admissible spherical representation and let be a spherical vector. Then is also spherical for , so there exists such that . Such defines a character of , and we cal the character of associated with the spherical representation . We proved that irreducible admissible representations are determined by their characters in Theorem . For spherical representation, stronger result holds - determines the representation.

We can do more. We can study the precise structure of in terms of simple generators and relations. For , let be the characteristic function of the set of all such that the ideal generated by in is . Also, let be the characteristic function of . ( is invertible) Then we have a nontrivial and simple relation, which might be familiar to you.

Now we have a natural question - which representations are spherical? First simple but nontrivial examples are principal series representations with unramified chracters.

Is there any other spherical representations? Obviously, there are simpler ones: 1-dimensional representations, which are just unramified quasicharacters of . We will show that these are all, i.e. there are no other spherical representations. For this, we need to know how acts on the spherical vector in the spherical principal series representation.

Using this with Theorem , we can prove that the only spherical representations are principal series representations and 1-dimensional representations.

We can also compute the action of intertwining operator on the spherical vector.

There are two special functions on associated with a spherical representation, called spherical Whittaker function and spherical function. Let’s study spherical Whittaker model first.

The integral absolutely converges if is dominant, i.e. . For general , we can define it as a limit which makes sense and defines a Whittaker functional.

We can compute the spherical Whittaker function explicitly. Note that we have for and , where is the central quasicharacter of . So it is sufficient to compute as runs through a set of coset representatives for , and by Iwasawa decomposition, it is sufficient to compute for .

Spherical function is defined as where are normalized spherical vectors so that . We can also compute this function explicitly. Note that is -biinvariant and , so we only need to compute its values on a coset representatives for . By Cartan decomposition, it is sufficient to compute for .

We can ask a different kind of question. When principal series representations are unitarizable? If the representation is induced from unitary data, then it is also unitary.

However, there may exist other principal series representations that are unitary but not induced from unitary characters.

So we want to know when is unitary. If we write with a unitary character and , Then where . So we are reduced to determine when is unitarizable.

These representations are called complementary series representation (recall that there is also complementary series representation for ). In summary, we have the following result.

Local zeta functions and local functional equations

In this section, we will define local zeta functions and prove local functional equations, which will be used to define and prove global functional equations of global automorphic -functions in Chapter 4. We will use some notations from Tate’s thesis, so you may have to read Chapter 4.1 first. To define local zeta functions,

First, we will show that Jacquet module controls the asymptotics of the functions in the Kirillov model of . This will allow us to define local zeta functions (we will define it as an integral of Kirillov model over , and the following results control the convergence).

Now we can completely understand what is as a Kirillov model.

Now we will see that we can control the asymptotic of functions in the Kirillov model of representations of near 0. The previous theorem tells us that if is a supercuspidal representation, then the functions in the Kirillov model vanishes near 0. We will also study the other two cases - principal series representations and special representations.

Now we can define local -function for given and a quasicharacter , and local zeta functions for (identified with the Kirillov model). The above theorems about asymptotes of Kirillov models will allow us to define local zeta functions.

The next theorem gives us local functional equations of local zeta integrals.

We call the meromorphic function (which does not depend on the choice of ) as a gamma factor. The next proposition shows that the gamma factors determine the representation .

There’s a simple relation between this (local) gamma factor for and , i.e. Tate gamma factors.

Global theory

Using local theories, now we can define -automorphic forms. We will glue local theories and interpret things in adélic language. Also, we will see how to interpret the classical modular forms and Maass forms in this way. Before we start, we will study -theory first, which is developed by Tate in his celebrated thesis. His thesis shows how powerful adélic languages are, and why this is the right way to study global things.

After that, we define the notion of automorphic forms and representations for , and define -functions attached to automorphic representations for , by generalizing Tate’s idea. Here we need Flath’s decomposition theorem and multiplicity one theorem.

Tate’s thesis

Tate’s thesis is a theory of -automorphic forms over a global field. In 1950s, Riemann proved that his famous Riemann zeta function has an analytic continuation and a functional equation, by using the theta function. Tate re-proved this fact, but in a completely different way. Tate’s idea is the following:

1. Develop Fourier theory on adéles , including Fourier transform and Fourier inversion formula.

2. Define adélic version of Hecke -functions and local & global zeta integrals. Prove functional equation for these zeta integrals.

3. Show that the local zeta integrals coincides with the local -functions for all but finitely many places.

4. Derive analytic continuation and functional equation for Hecke -functions from corresponding local statements. Also, Euler product becomes simply a factorization of global integral according to the product structure of .

This gives a natural way to get the global result from local results, and we will develop -theory via similar way.

Let be a global field (number field or function field over a finite field), and let be its adéle ring, i.e. the restricted product where runs over the set of places of and is a completion of with respect to . For non-archimedean , is a ring of integer of . This is a locally compact abelian group and we have Haar measure on it, which is both left and right invariant. can be embedded into diagonally, and the quotient is compact. It is known that we can always find a nontrivial additive character on that is trivial on . Also, any continuous character of has the form for some , and gives an isomorphism . Let be a ring of finite adéles, which is a restricted product of ’s for non-archimedean .

Now we want to define Fourier transform as where is a Haar measure on . However, there are two problems with this definition.

First, the integral does not converge for some . To fix this, we consider smaller but dense subspace, which is the space of Schwartz functions.

For Schwartz functions, the integral absolutely converges and the problem is resolved. However, there’s one more problem. We want that the Fourier inversion formula holds, so that for all . To do this, Haar measures on and its dual should be compatible in some sense. We saw that by fixing a nontrivial additive character , and this gives a unique normalization of the Haar measure for which the Fourier inversion formula holds. Such measure is called self-dual Haar measure. If ’s are local self-dual measures for each place , then is a self-dual measure of , and same thing holds for on .

We can also think Dirichlet characters in adélic setting. Such characters are called Hecke characters.

First, for all but finitely many , local components are unramified:

Now the following proposition shows that finite order Hecke characters and Dirichlet characters are just same things, at least for .

This also holds for general global fields. This proposition will be used later to show that the classicial Dirichlet -function (or Hecke -function for general number fields) is same as the adélic version of it.

If , then and so we can assume that is of finite order. We will define for later.

We will show that the above factorization of zeta integral makes sense, i.e. it converges for . Also, we will show functional equations of local zeta integrals, which automatically gives the functional equation for the global zeta integral.

By combining all the previous results, we get the analytic continuation and functional equation of a Hecke -function.

We will complete the -function by defining for , too. In this case, the functional equation will contain some extra factors called factors. If is non-archimedean (so that is ramified), then define . If is real, for some , and we define . Finally, for complex , for some and (since it is unitary). (Here is the square of the usual complex norm.) Then we put

We say that a nonzero function is of exponential type if for some and . We will show that we can choose appropriate so that the quotient of local zeta integrals by local -functions became a function of exponential type.

The following proposition describes so-called (local) factor (or root numbers), which is an extra factor for the completed -function.

Now we define where the product is over all places of . The next theorem show that the completed -function has meromorphic continuation with the functional equation that contains . Also, we show that doesn’t depend on the choice of , although the local factors do.

Definition of automorphic forms and representations

Now we will develop similar theory for . Before doing this, we first define automorphic forms of , which is a generalization of both modular forms and Maass forms, and then define automorphic representations which use adélic language.

Let and let be a discrete subgroup that contains and cofinite, i.e. has a finite volume. But we do not assume that is compact, so that we will allow discrete subgroups like (congruence subgroups) that has cusps. Let , be a character of , and let be a character of the center of . Here we assume that all the characters are unitary.

We also define cusp forms. Assume that is a cusp of so that contains an element of the form . We say that is cuspidal at if either or If is an arbitrary cusp, we can find such that . Then is an element in with and . We say that is cuspidal at if is cuspidal at .

For example, modular forms and Maass forms are automorphic forms with some additional conditions, such as being holomorphic or being an eigenvector of Laplacian operator. Also, we know that modular forms can be regarded as a Maass form: if is a weight modular form, then is a Maass form with the eigenvalue .

Now we relate these classical automorphic forms to -modules in the previous theorem. Let be a Maass form of weight . Define a function as One can check that , where is the character of that is trivial on the connected component of the identity and agrees with on . Since is an eigenfunction of , is an eigenfunction of and it is -finite. Also, the function satisfies the equation , which implies that is also -finite. Growth condition of is equivalent to that of , so that generates an admissible -module. The effects of on are transferred into the effects of on .

We are also interested in the decomposition of (right) regular representation of and . As before, we have the corresponding action of Hecke algebra given by for . is an operator on leaving invariant, and is a unitary representation on both spaces. Also, we can rewrite the above equation as an integral over : where Our aim is to prove that is a compact operator, so that the space decomposes into a Hilbert space direct sum of irreducible invariant subspaces. To prove that, we need a notion of Siegel sets. For , we denote by the Siegel set of such that and . Also, we denote by the set of such that .

We can lift these Siegel sets to subsets of under the map . Let be the preimages of in . Also, we denote . Then the previous proposition also holds for fundamental domains of and .

We can also consider the subspace of automorphic forms which are cuspidal, -eigenspace of , and -isotypic. The following theorem shows that this space is finite dimensional, and all the irreducible admissible unitary representations of occur in with finite multiplicity.

Now, we are going to transfer everything in terms of adélic setting. In particular, we will define automorphic forms and representations of . This is a modern point of view and this is also a natural way to study in philosophy of local-global principle - think about Tate’s thesis. (Actually, Tate’s thesis is exactly about the theory of automorphic forms.)

Before we define them, we will prove the adélic version of the Theorem , which can be prove by using the adélic version of the Proposition . Ideas are almost same and we only need to define everything properly in terms of adéles.

To prove the decomposition theorem of , we need some well-known properties of . We use the following propositions in the proof of the Proposition , but we will not prove these theorems. See Theorem 3.3.1, Proposition 3.3.1, and Proposition 3.3.2 in (bu?).

Note that this holds for any and a number field . Now we can prove our main proposition for the decomposition theorem.

This with the spectral theorem prove the following decomposition theorem.

Now we define automorphic forms and representations of . Automorphic forms of are functions on that satisfies the transformation law, the -finiteness and -finiteness condition, and the growth condition.

Note that does not act on the space since -finiteness is not preserved by right translation by elements of for archimedean . However, it is still a representation of and also -module, where and . This motivates the following definition.

In Section 4.6, we will explain how to attach an automorphic representation by using the classical automorphic forms, for example, modular forms.

We can also define the notion of an admissible representation of .

In Section 4.6, we will see that this is equivalent to a representation of the global Hecke algebra. The following theorem shows that any irreducible subrepresentation of induces an irreducible admissible representation of .

Flath’s decomposition theorem

We can ask a simple but hard question: how to construct a nontrivial example of (irreducible admissible) representation of , and how can we study? There’s one possible and natural way to do it by glueing local representations. To be more specific, first we define restricted tensor product of representations.

For each , we have natural injective maps and we denote the image of under this map as where for . Such element is called a pure tensor. We can consider as a vector space spanned by pure tensors. One can check that changing finite number of does not change the restricted tensor product.

Using this, we can also define a restricted tensor product of representations.

In this note, we assume that is a set of places of some global field , be maximal compact group of and or . The following lemma shows that a finite dimensional representation of the maximal compact subgroup can be decomposed into local representations.

Now Flath’s decomposition theorem (tensor product theorem) says that any irreducible admissible representation of decomposes as a restricted product of local irreducible representations, where almost all of them are spherical.

We will not give a complete proof here. The proof uses properties of so-called idempotented algebras. One can use theory of idempotented algebras and apply it to Hecke algebras. For a non-archimedean place , is a convolution algebra where became a spherical idempotent element of when the Haar measure is normalized so that the volume of is 1. (These are commutative by the Cartan decomposition theorem). For an archimedean place , define as a convolution algebra of compactly supported distributions on that are supported in and -finite under the left and right translations. (The support of contains . itself contains both and , and every element of has the form with and .) Then , which is -module, became a restricted tensor product of -modules (with respect to spherical Hecke algebras ) which are just representations of ).

By using the decomposition theorem, we can define the contragredient representation of an irreducible admissible representation of as . We already defined contragredient representation of for non-archimedean in Section 3.1, and we can also define the contragredient representation of given -module as a -module by for and . Then we can show the global analogue of the Theorem , which almost directly follows from the local results. Note that archimedean analogue of the theorem is also true, but we will not prove here. One can prove it by using the classification of irreducible admissible -modules of .

Whittaker models and multiplicity one

In this section, we will prove that irreducible representation of is determined by all but finitely many local components. More precisely:

We will only show this for . To prove this, we will construct global version of Whittaker model (by glueing local Whittaker models) and prove existence and uniqueness.

First, we will prove the result for function field, so that we can ignore archimedean places. Let be a nontrivial additive character of and let be an irreducible admissible representation of .

The following theorem proves that Whittaker functional is unique and always decomposes as local Whittaker functionals.

To prove global uniqueness for number fields, we have to prove uniqueness theorem for archimedean places, too. We already proved for -modules of and the same statement is also true for . To unite both archimedean and non-archimedean cases, we will consider as -module for , where is a local field. (We’ve defined in the previous section.) For such , let be a simple admissible -module and denote the action by .

In terms of this formulation, we can describe the uniqueness of local Whittaker models as follows.

Global Whittaker model is almost the same as the local definition. Let be a global field, let be the set of all places of , and let be the restricted tensor product of the with respect to the spherical idempotents (see the previous section). Let be a nontrivial additive character on .

To prove the uniqueness theorem, we need one more proposition.

When is an automorphic cuspidal representation of , then we can explicitly construct Whittaker models in terms of integral and prove existence. This is not true in general - for example, there’s no Whittaker models for .

Using the existence and uniqueness of global Whittaker model, we can prove the multiplicity one theorem for .

Automorphic -functions for

The multiplicity one theorem is important as itself, but it is also important because we can construct automorphic -functions, which are -functions attached to cuspidal automorphic representations. This is a generalization of Tate’s thesis for automorphic forms.

In the previous section, we defined local -functions for an irreducible admissible representation of and a quasicharacter . Using this, we can define partial -function as follows.

Our aim is to prove the functional equation of , by using the existence and uniqueness of the global Whittaker model. As we did in the Tate’s thesis, we first define a global zeta integral.

By rapid decay of , we can show that the global zeta integral converges for any . Indeed, is rapidly decreasing as , i.e. for any there exists a constant such that for sufficiently large . Also, since is automorphic, so also rapidly decreases as . This implies that the global zeta integral (of the first form, integral over ) absolutely converges for any . (Recall that local zeta integrals converge for sufficiently large .) However, the second form (integral over ) does not absolutely converges for all , but for sufficiently large .

By simple transformation, we can prove a functional equation of global zeta integral.

Following proposition shows that the local zeta integral coincides with local -functions for unramified places.

Now, we are ready to prove the global functional equation of -function.

To complete the -function, we want to define local -functions for ramified places . For any place , we should have holomorphic for all , and if we define then is a function of exponential type.

When is a principal series representation, we define Where the -factors on the RHS are as defined in section 4.1. In this case, we have If is non-archimedean and is a special representation (so that ), then we have and For other cases (such as supercuspidal representation on non-archimedean place), we put . In these cases, we have .

Adelization of classical automorphic forms

In this last section, we will study how to get automorphic representations from classical automorphic forms such as modular forms and Maass forms. For a given modular form, we can lift a function as a function on the adéle group , then consider a -module generated by the function. This is an irreducible admissible representation of by Theorem . We will describe this procedure more rigorously and prove that the associated representation is automorphic when the original modular form is a Hecke eigenfunction.

Let be a modular form or Maass form of weight on with a character . We already saw that the function defined as is of moderate growth, an eigenfunction of , and for . Here is a Dirichlet character modulo (not necessarily primitive).

To adélize as a function on , we also need to adélize as a character of . In Proposition , we saw that there’s 1-1 correspondence between Dirichlet characters and characters of . So we have a character of corresponds to , so that for and . Also, is unramified for and is trivial on the subgroup of consisting of elements . is trivial on .

Now we define a character of by where is a set of non-archimedean places dividing . By strong approximation theorem, we can write any by with . Then we define as associated function on . This function is well-defined: this follows from the equation for coprime to . This is an automorphic form with a central quasicharacter , which can be shown by using

We can also extend classical Hecke operators to adélic setting. For each prime , the corresponding adélized Hecke operator will be the operator in the local spherical Hecke algebra , which is defined in Section 3.8.

First, we define Hecke operator for automorphic forms on . Let and let be the localization of at the prime in , so that iff . We can trivially extend the Dirichlet character to . If we put , then we have a right action of on functions on by Then for , we can define the Hecke operator as where is a complete set of coset representatives for . Especially we put for . Now, the following theorem shows that every Hecke eigenform gives rise to an automorphic representation.

We can also obtain theorems for classical automorphic forms by adélizing it and use tools that we proved for automorphic representations. For example, the global multiplicity one theorem implies that if we know almost all Fourier coefficients of a modular form, then it is uniquely determined. This theorem also holds for Maass wave forms.

Note that this is not true for general congruence groups with characters - we also need to assume that are newforms.